Chapter 20. Finite Element Method (FEM) Introduction

Transcription

1 Chapter 20. Finite Element Method (FEM) Introduction The objective of this chapter is to explain basic terms of the particular field of problems and practical application of GEO 5 FEM program to solve some geotechnical problems. GEO 5 FEM program allows for the modelling of various types of problems and analyses. The text below explains basic terms and general procedures in a more detailed way individual analysis modules are described in other chapters. From the problem type aspect, GEO 5 FEM program distinguishes two basic cases: a planar problem: this analysis module is suitable for solving linear structures (a tunnel, embankment, open cut, dam etc.), for which it applies that their longitudinal dimension is by an order of magnitude larger than the lateral dimensions of the area being solved. In such a case it is proper to relate the analysis to 1 rm of the structure and solve the problem under the assumption of plane-strain deformation. Deformations in planes parallel to the longitudinal axis of the structure can be disregarded in this case. Thus the development of deformations and stresses in the massif is taken into consideration only in the plane perpendicular to the longitudinal axis and, as a result of lateral contraction, also the normal stress in the longitudinal direction. Regarding beam elements, it is the case of solving a 1m wide plate strip (for more details visit Help F1). axial symmetry: this analysis module is suitable for solving rotationally symmetric problems. This assumption must be satisfied by both the geometrical arrangement of the structure and loading. A suitable example is the analysis of a vertically loaded single pile, circular excavation or pumping groundwater from a circular borehole. Similarly to the case of the plane-strain deformation, this is the case of a generally three-dimensional problem, which, however, can be again converted into the planar problem solution. The solution is then related to 1 rad of the x (r) -radius arch. The axis of symmetry always represents the x (r) coordinate origin.

2 Shear components of deformation in the direction of rotation can be disregarded. The evolution of a circumferential normal component of stress and deformation is also taken into consideration, in addition to the components of stress and deformation in the cross-sectional plane (for more details visit Help F1). From the type of analysis point of view, the program allows the following cases to be solved using individual modules: stress: it serves to solve basic geotechnical problems in ground environment and a rock mass (e.g. for determining the vertical or horizontal geostatic stress, pore pressure, deformations, volumetric changes and sub-grade deformations, as well as analysing internal forces along a diaphragm wall structure length (height) etc.). steady flow: it assumes a zero change in the saturation degree with time; individual construction stages are completely independent of each other (in contrast with the unsteady flow). unsteady flow: this analysis module allows for determination of the evolution of pore pressures (the total head) and the current degree of saturation with time possible. In this case, the analysis methodology is similar to that of the stress analysis. slope stability: during the analysis, this program reduces the input values of the internal friction angle ef or the soil cohesion c ef and seeks the onset of failure associated with the evolution of the critical region of localized plastic deformation. The result is the factor of safety corresponding to classical slope stability analysis methods. The definition and input of a model in this regime is completely identical with that of the Stress module. tunnels: this module allows for performing the analysis of underground excavation (modelling of the 3D effect of excavation attributed to the New Austrian Tunnelling Method), accounting for degradation of beams, temperature-induced loads acting on beams, swelling-induced loads acting within specified regions and monitoring of results.

3 Settings frame GEO 5 FEM program incorporates the possibility of so-called advanced specification, where it is possible to define in more detail complementary input parameters of soils for individual material models, to develop mixed meshes with multi-nodal elements and visualise larger numbers of output variables. Note: The standard setting assumes drained boundary conditions. In such the case the analysis assumes steady-state conditions, where the skeleton deformation has no influence on the development of pore pressures. In such a case pressures have only the character of external loads and do not vary during the course of the analysis. In the case of undrained conditions, where the entire boundary of the particular area behaves as being fully impermeable, we solve the opposite case, i.e. the fully coupled problem of the skeleton deformations and pore pressures under the assumption that all changes are immediate and the influence of time does not take effect. The method of determining the initial stress (1 st in the program as: construction stage) is assumed geostatic stress: The standard method for the analysis of the vertical geostatic stress is based on the following relationship: z n i1 h i i kpa where: i bulk density of soil in i th layer, h i thickness of i th layer.

4 K 0 procedure: It is used in the cases where the user needs to define another initial lateral stress. For example, the actual lateral stress in overconsolidated soils can be significantly higher than that in normally consolidated soils (for more details visit Help F1). The lateral pressure coefficient K 0 is set as a soil parameter. If this parameter is not specified, it is provided by: K 0 1 where: Poisson s ratio. Input parameters of soils also depend on the selected material model for stress or flow analysis, respectively. The most important input parameters for stress analysis comprise the modulus of elasticity E and Poisson s ratio (to be specified for all models), and for non-linear models, the angle of internal friction ef and the soil cohesion c ef. The steady-state condition after the redistribution of pore pressures is assumed in the program and effective parameters of shear strength of soils or rocks are therefore used during the analysis (for more details visit Help F1). The selection of the material model and subsequent setting of soil parameters belong among the most important and at the same time most problematic tasks when the FEM modelling of a structure is being carried out. Material models try to plausibly describe the behaviour of soil or rock). They can be divided into two basic groups, namely linear and non-linear models. Note: The correct selection of the material model is absolutely necessary for giving a true picture of the real behaviour of the structure. Non-linear models are necessary for the majority of structures (e.g. the analysis of a sheeting wall structure using a linear model of soil will give completely unrealistic results), but the application of linear models can be in many cases very proper, simplifying the entire analysis. A simplified procedure recommended for the modelling of problems using the FEM is presented in Help F1). Linear models give relatively fast, but not very accurate, assessment of the real material behaviour. They can be used in the cases where we are interested only in the stress or deformations of the ground mass, not in the area and mode of the potential failure.

5 They can also be used in the cases where only a local failure develops, having no fundamental influence on the development of a global failure, but which may result into the premature termination of the analysis in the program (for more details visit Help F1). The group of linear models comprises: elastic model: It uses conversion relationships between the stress and deformation given by Hooke s law (within the linear elasticity scope). modified elastic model: It makes incorporating the influence of surcharge loading or unloading into the analysis possible, using the secant modulus E def and the unloading/reloading modulus E ur. Although, if we try to obtain a plausible description of the ground mass behaviour or if we are interested in the distribution location of areas of potential failures, it is necessary to adopt non-linear models. Basic non-linear models can be again divided into two groups. The first group of models is based on the classical Coulomb failure condition. It comprises the Drucker-Prager, Mohr-Coulomb and Modified Mohr-Coulomb models. Using these models, it is also possible to model hardening or softening of soils. A common feature of these models lies in the unlimited elastic deformation under the assumption of geostatic stress (for more details visit Help F1). The second group of material models, which are based on the notion of the critical state of soil, is represented by the Modified Cam-clay, Generalized Cam-clay and Hypoplastic clay models. These models provide a significantly better picture of the nonlinear response of soil to external loading. Individual material models differ not only in their parameters, but also in the assumptions. The boundary between the linear (elastic) and non-linear (plastic) response of the material is formed by the surface of plasticity. The mathematical expression of the surface of plasticity represents a certain failure condition (plasticity function). Exceeding this condition leads to the development of permanent (irreversible) plastic strains.

6 Note: Apart from the basic material parameters adopted for linear models, the non-linear models require introducing certain strength-related characteristics of soils, which are necessary for the formulation of yield conditions. The onset of evolution plastic strain depends on the value of angle of internal friction and cohesion c. The dilatancy angle controls informs magnitude of the plastic volumetric strain (for more details visit Help F1). The selection of a material model suitable for the analysis of geotechnical structures adheres first of all to the character of the soil/rock environment. In the Finite Element Method-based process of comprehensive modelling of more complex problems, the selection of the numerical model represents an absolutely essential influence on specifying the input data and assessing the analysis results. Working with interface, model dimensions: The description of the work with the specification of individual interfaces is in a more detailed way presented in the program help (see F1). The input data essential for numerical analyses using the FEM are the so-called world coordinates (specifying the magnitude of the area being solved), where it is necessary, especially for stability analyses, to secure sufficient neighbourhood (the interface width) to obtain meaningful results. Note: The depth of the FE mesh is also very important. It is possible to visualise the end of the mesh as an incompressible sub-grade. If there is no incompressible sub-grade specified for the particular geological profile, it is possible to assume that internal forces will disappear at a certain distance from the loading spot or from the contact of the structure with the sub-grade; no deformation will therefore develop. The world boundaries of the problem being solved are subsequently defined to be at this distance (for more details visit Help F1). The interface can be imported from other programs of the GEO 5 system via the clipboard. The program further allows the import and export of interfaces in *DXF format and the import of interfaces in gint format. To simplify the specification of interface points (geometry) it is possible to use the so-called interface corrector (for more details visit Help F1).

7 Mesh generation: The successful generation of a mesh is the last item of the process of setting the structure topology (interfaces between soil layers, lines of structures, parameters of soils and rocks, contacts etc.). Individual construction stages are modelled and analysed subsequently. When the mesh is being generated, the program automatically generates also the standard boundary conditions. The standard setting of the boundary conditions comprises: smooth pin at the mesh nodes found along the horizontal bottom edge, sliding pin at the mesh nodes along the left-hand and right-hand vertical edges. The GEO 5 FEM program has a built-in automatic corrector of the specified geometry. This means that the program itself finds intersections of lines and all closed areas and develops an adequate model before the generation of the finite element mesh (for more details visit Help F1). The newly developed areas can be subsequently removed from the analysis or new soil can be assigned to them. The user will become aware of the main advantages of this system particularly when analysing tunnels and braced structures. Specifying even a very complicated structure becomes a very simple and fast matter (for more details visit Help F1). The correctly generated finite element mesh is the basic condition for obtaining results regarding the real behaviour of the structure. GEO 5 FEM program has a built-in automatic mesh generator, which significantly simplifies this task. Despite this fact it is necessary to adhere to certain rules: The denser the mesh the more accurate results. On the other hand, the problem analysis becomes significantly slower. The objective therefore is to find the optimum density of the mesh. It depends both on the user experience and the finite element used.

8 The mesh of elements should be sufficiently dense, especially in the area where large stress gradients can be expected (spot support, sharp corners, underground excavations etc.). It is perform mesh refinement around individual points or lines. It is necessary for the density-refining radius to be at least 3 to 5 times larger than the density in the density refining centre and both values at the points (the density and radius) to be in a reasonable proportion to the mesh density prescribed for the surrounding area. In this way smooth transition between the areas with different density will be secured. Note: Singular lines must be dealt with similarly. In the cases of more complicated problems it is advisable to carry out a preliminary analysis of a course finite element mesh and on the basis of the analysis of generated results, to carry out a local refinement of the mesh density (for more details visit Help F1). The program uses six-node triangular elements with automatic mesh smoothing by default (for more details visit Help F1). Construction stage: When the structure topology specification and the finite element mesh generation are finished, the analyses are then carried out for individual construction stages. The construction stages correspond to the gradual building of the construction, and the correct specification and sequence are very important. The analysis of each stage (with the exception of stability analyses) is based on the results of the preceding stage. Information about individual construction objects and their properties is maintained between the construction stages. The rule of properties inheritance is applied between the construction stages when a stage is being edited or specified (for more details visit Help F1). The first construction stage (the analysis of the primary geostatic stress) represents the initial state of the massif before the construction commencement. For that reason the analysis result refers to the stress in soil or rock mass, not to deformations.

9 Note: A principal problem of the FEM analysis usually lies in the non-convergence of some of the construction stages. If the results of one phase are not available, the analysis of the subsequent stage is impossible. As for the correct modelling, the program authors recommend to follow a recommended procedure for the modelling of construction process (for more details visit Help F1). Analysis settings and description: During the course of the analysis the program tries to find (iterate) such a solution at which the equilibrium conditions in the massif are met for the specified boundary conditions. The iteration process and the analysis convergence can be followed on the screen (for more details visit Help F1). The analysis can be interrupted at any time; in such a case the results are available for the last successful convergence of the loading increment. Correct results can be obtained when the 100 per cent loading is reached. Nevertheless, it may happen that the analysis passes off only up to a certain percentage of the loading, which means that the program has not succeeded in finding the correct solution and the analysis does not converge (for more details visit Help F1). In such the case it is possible to change the standard analysis parameters setting or to carry out some modifications of the model: increasing the structure stiffness, reducing the applied loads, reduce the soil excavation area, improving soil characteristics, changing the material model of soils in plasticity locations, adding beam reinforcing elements or tensional elements, adding boundary conditions, changing the iteration course in the analysis settings (e.g. increasing the number of iterations). Note: The visualisation of plastic strains showing critical locations with the anticipated development of failure surfaces can provide the advice why the analysis does not converge (for more details visit Help F1).

10 The program provides a default set of analysis parameters ensuring sufficient accuracy and speed. However, an experienced user may want to change certain parameters or test their influence on the analysis accuracy (for more details visit Help F1). Note: The authors of the program recommend that any changes in the setting of analysis parameters should be approached very carefully, after thoroughly studying the problems. Incorrectly selected settings may cause not only incorrect iteration of the solution and slowing the analysis down, but also inaccurate results (for more details visit Help F1). It is possible in the analysis stage to change the following parameters in the Settings window: the solving method (and its settings): Newton-Rapshon method NRM, Arc-length method ALM. the stiffness matrix: the initial stress method; the full or modified NRM. the initial computation step: the ratio between the load in the particular loading step and the prescribed overall load (the standard setting value is 25 per cent of the overall load). the maximum number of iterations: for achieving equilibrium within the framework of the given loading increment. the convergence criterion: the setting of the error tolerance (changes in the vector standard) for displacement, unbalanced forces and internal energy. the line search method: the determination of the weight coefficient for meeting the equilibrium condition, resulting in the acceleration or dampening of the computation process. plasticity: the setting of the tolerance for the error in the return to the yield surface, expressing the accuracy required for meeting the yield condition.

11 Note: Individual analysis settings, including basic equations for meeting the equilibrium conditions or yield conditions, are described in more detail in the program Help (for more details visit Help F1). Visualisation and interpretation of results: The visualisation and interpretation of results is one of the most important parts of the program. GEO 5 FEM program allows several basic types of graphical outputs: the drawing of the deformed structure planar representation of quantities in the ground mass (it is possible to represent absolute values or values compared with another construction stage) internal forces in beams and on contacts forces in anchors and reactions settlement trough tilted cross-sections or vectors and directions of the quantities Note: The program uses certain coordinate systems for the representation of results. All outputs and selected results can be printed in the analysis protocol (for more details visit Help F1). Some results cannot be drawn concurrently because of the necessity for transparency and comprehensibility. It is, for example, impossible to draw a deformed structure and, at the same time, the distribution of internal forces along a beam. It is always necessary to choose only one variant. The program gives notice at the bottom of the dialogue window in the case of setting inadmissible combinations of outputs. The program makes the setting of an arbitrary quantity of point monitors and line monitors possible at any place of the structure or outside the structure. Monitors have several functions: visualising the values of quantities at a particular point (the point monitor), visualising the values of the difference at a distance between two points compared with the previous construction stage N N, is the number of the construction stage (the line monitor). d, or compared with the setting phase where

12 List of chapters relating to the FEM: Chapter 20: Finite Element Method (FEM) Introduction. Chapter 21: Terrain settlement analysis. Chapter 22: Settlement of a circular silo foundation. Chapter 23: Collector lining analysis. Chapter 24: Numerical solution to a sheeting wall structure. Chapter 25: Slope stability assessment. Chapter 26: Numerical modelling of tunnel excavation adopting the NATM method. Chapter 27: Steady water flow analysis Body of the embankment. Chapter 28: Steady water flow analysis Flowing in and out of a well. Chapter 29: Unsteady water flow analysis Filling of a dam reservoir.